ON SOLUTE TRANSPORT IN OSCILLATORY FLOW THROUGH AN ANNULAR PIPE WITH A REACTIVE WALL AND ITS APPLICATION TO A CATHETERIZED ARTERY by B. S. MAZUMDER and KAJAL KUMAR MONDAL (Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700 108, India) [Received 19 December 2002. Revises 23 April and 25 November 2004] Summary Longitudinal dispersion of passive tracer molecules released in a pulsatile flow through an annular pipe with a reactive outer wall is studied by employing the method of integral moments. It is shown how the spreading of tracers is influenced by the shear flow due to a periodic pressure pulsation in the pipe and a first-order reaction at the wall. The behaviour of the dispersion coefficient due to variation of the aspect ratio (the ratio of the inner radius to the outer radius of the annular pipe) for periodic flow with and without a non-zero mean is examined. It is shown that an increase of aspect ratio, Womersley parameter or reaction at the outer wall of the annular pipe inhibits the dispersion of tracers. The axial distribution of the mean concentration is approximated using a Hermite polynomial representation from the first four central moments for a range of different aspect ratios and frequencies of the pressure pulsation. It is interesting to note that for low frequencies of pulsation, an increase of aspect ratio leads to a significant effect on the concentration distribution, whereas for large frequencies, this effect tends to diminish. The results of this study are of great importance in understanding the dispersion process in a catheterized artery with a reactive arterial wall. 1. Introduction The study of longitudinal dispersion of a tracer in a straight tube has a wide range of applications in the fields of chemical, environmental and biomedical engineering. The dispersion of diffusing solute in a fluid flowing through a circular impermeable tube was first described by Taylor (1) and subsequently extended by Aris (2) for pipe Poiseuille flow using the method of moments. They confirmed that after a sufficiently long time when the tracer was completely mixed across the tube, any localized initial configuration of the tracer material evolved to a Gaussian distribution moving with the mean speed of the flow. Using his method of moments, Aris (3) analysed the longitudinal dispersion of solute in an oscillatory flow of a viscous incompressible fluid under a periodic pressure gradient. However, his analysis of the dispersion coefficient was restricted to asymptotically large times after the injection of the solute. Watson (4) studied the mass transfer of a diffusing substance in oscillatory flow through a pipe. Grotberg (5) analysed oscillatory viscous flow in a tapered channel under conditions of fixed Stokes volume by developing a lubrication theory and Gaver and Grotberg (6) verified experimentally that both theoretical and experimental results show a bi-directional drift for all frequencies depending on the value of the Womersley parameter. Rao and Deshikachar (7) explored the generalized dispersion model proposed by Gill and Sankarasubramanian (8) to study the dispersion of a diffusing solute in an annular pipe and evaluated the dispersion coefficient for Q. Jl Mech. Appl. Math, Vol. 58. No. 3 c  The Author 2005. Published by Oxford University Press 2005; all rights reserved. For Permissions, please email: journals.permissions@oupjournals.org doi:10.1093/qjmam/hbi009 Downloaded from https://academic.oup.com/qjmam/article-abstract/58/3/349/1860946 by guest on 16 June 2020